Algebra and Applications 2

Let s be a subforest of a rooted tree t . Denote by t / s the tree obtained by contracting each connected component of s onto a vertex. We turn ℋ into a bialgebra by defining a coproduct Δ : ℋ → ℋ ⊗ ℋ on each tree by :

[1.88]

where the sum runs over all possible subforests (including the unit • and the full subforest t ). As usual we extend the coproduct Δ multiplicatively onto . In fact, coassociativity is easily verified. This makes ℋ := ⊕ n ≥ 0ℋ na connected graded bialgebra, hence a Hopf algebra, where the grading is defined in terms of the number of edges. The antipode S : ℋ → ℋ is given (recursively with respect to the number of edges) by one of the two following formulae:

[1.89]

[1.90]

It turns out that ℋ CKis left comodule-bialgebra over ℋ (Calaque et al . 2011; Manchon and Saidi 2011), in the sense that the following diagram commutes:

Here, the coaction Φ : ℋ CK→ ℋ ⊗ ℋ CKis the algebra morphism given by Φ( 1) = • ⊗ 1and Φ( t ) = Δ ℋ( t ) for any nonempty tree t . As a result, the group of characters of ℋ acts on the group of characters of ℋ CKby automorphisms.

Let M be a differentiable manifold, and let ▽ be the covariant derivation operator associated with a connection on the tangent bundle TM . The covariant derivation is a bilinear operator on vector fields (i.e. two sections of the tangent bundle): ( X, Y ) ↦ ▽ XY , such that the following axioms are fulfilled:

The torsion of the connection is defined by:

[1.91]

and the curvature tensor is defined by:

[1.92]

The connection is flat if the curvature R vanishes identically, and torsion-free if . The following crucial observation by Matsushima (1968, Lemma 1) is an immediate consequence of equation [1.43]:

PROPOSITION 1.14.– For any smooth manifold M endowed with aflat torsion-free connection ▽, the space χ ( M ) of vector fields is a left pre-Lie algebra, with pre-Lie product given by :

[1.93]

Note that on M = ℝ n, endowed with its canonical flat torsion-free connection, the pre-Lie product is given by:

[1.94]

Cayley (1857) discovered a link between rooted trees and vector fields on the manifold ℝ n, endowed with its natural flat torsion free connection, which can be described in modern terms as follows: let be the free pre-Lie algebra on the space of vector fields on ℝ n. A basis of is given by rooted trees with vertices decorated by some basis of χ (ℝ n). There is a unique pre-Lie algebra morphism , the Cayley map , such that for any vector field X .

PROPOSITION 1.15.– For any rooted tree t, with each vertex v being decorated by a vector field Xv, the vector field is given at x ∈ ℝ n by the following recursive procedure (Hairer et al. 2002): if the decorated tree t is obtained by grafting all of its branches tk on the root r decorated by the vector field , that is, ifit writes , then :

[1.95]

[1.96]

where stands for the k th differential of fi .

PROOF. – From equation [1.94], for any vector field X and any other vector field :

[1.97]

In other words, X ⊳ Y is the derivative of Y along the vector field X , where Y is viewed as a C ∞map from ℝ nto ℝ n. We prove the result by induction on the number k of branches: for k = 1, we check:

Now, we can compute, using the Leibniz rule and the induction hypothesis (we drop the point x ∈ ℝ nwhere the vector fields are evaluated):